An LIL Version of L = λW

This paper establishes a law-of-the-iterated-logarithm (LIL) version of the fundamental
queueing formula L = λW: Under regularity conditions, the continuous-time arrival counting
process and queue-length process jointly obey an LIL when the discrete-time sequence of
interanival times and waiting times jointly obey an LIL, and the limit sets are related. The
standard relation L = λW appears as a corollary. LILs for inverse processes and random sums
are also established, which are of general probabilistic interest because the usual independence,
identical-distribution and moment assumptions are not made. Moreover, an LIL for regenerative
processes is established, which can be used to obtain the other LILs.